Pricing Asian Options under a Hyper-Exponential Jump

Diffusion Model

Ning Cai1 and S. G. Kou2

1Room 5521, Department of IELM, HKUST2313 Mudd Building, Department of IEOR, Columbia University

ningcai@ust.hk and sk75@columbia.edu

June 2011

Abstract

We obtain a closed-form solution for the double-Laplace transform of Asian optionsunder the hyper-exponential jump diffusion model (HEM). Similar results are only availablepreviously in the special case of the Black-Scholes model (BSM). Even in the case of theBSM, our approach is simpler as we essentially use only Ito’s formula and do not needmore advanced results such as those of Bessel processes and Lamperti’s representation. Asa by-product we also show that a well-known recursion relating to Asian options has aunique solution in a probabilistic sense. The double-Laplace transform can be invertednumerically via a two-sided Euler inversion algorithm. Numerical results indicate that ourpricing method is fast, stable, and accurate, and performs well even in the case of lowvolatilities.

Subject classifications: Finance: asset pricing. Probability: stochastic model applica-tions.

Area of review: Financial engineering.

1 Introduction

Asian options (or average options), whose payoffs depend on the average of the underlying asset

price over a pre-specified time period, are among the most popular path-dependent options

traded in both exchanges and over-the-counter markets. A main difficulty in pricing Asian

options is that the distribution of the average price may not be available analytically.

There is a large body of literature on Asian options under the Black-Scholes model (BSM).

For example, approaches based on partial differential equations were given in Ingersoll [26],

Rogers and Shi [38], Lewis [31], Dubois and Lelievre [18], Zhang [51, 52]; Monte Carlo simulation

techniques were discussed in Broadie and Glasserman [7], Glasserman [24] and Lapeyre and

1

Temam [29]; analytical approximations were derived in Turnbull and Wakeman [46], Milevsky

and Posner [35] and Ju [27]; lower and upper bounds were given in Curran [16], Henderson et

al. [25], and Thompson [45]. Previous results that are related to ours are: (i) Linetsky [32]

derived an elegant series expansion for Asian options via a one-dimensional affine diffusion.

(ii) Vecer [47] obtained a one-dimensional partial differential equation (PDE) for Asian options

which can be solved numerically in stable ways. (iii) Based on Bessel processes and Lamperti’s

representation, in a celebrated paper Geman and Yor [23] provided an analytical solution of a

single-Laplace transform of the Asian option price with respect to the maturity; see also Yor

[50], Matsumoto and Yor [33, 34], Carr and Schroder [13] and Schroder [40]. Significant progress

has been made for the inversion of the single-Laplace transform in Shaw [41, 43]. Dewynne and

Shaw [17] gave a simple derivation of the single-Laplace transform, and provided a matched

asymptotic expansion, which performs well for extremely low volatilities. (iv) Dufresne [19, 20]

obtained many interesting results including a Laguerre series expansion for both Asian and

reciprocal Asian options. (v) Double-Laplace and Fourier-Laplace transforms were proposed in

Fu et al. [21] and Fusai [22], respectively. For the differences between their methods and ours,

see Section 2 and the online supplement (Section 3).

All the papers discussed above are within the Black-Scholes framework. There are only few

papers for alternative models with jumps. For example, Albrecher [2], Albrecher and Predota

[4] and Albrecher et al. [3] derived bounds and approximations for Asian options under certain

exponential Levy models; Carmona et al. [11] derived some theoretical representations for Asian

options under some special Levy processes; Vecer and Xu [49] gave some representations for

Asian options under semi-martingale models via partial integro-differential equations; Bayraktar

and Xing [6] proposed a numerical approach to Asian options for jump diffusions by constructing

a sequence of converging functions.

In this paper we study the pricing of Asian options under the hyper-exponential jump

diffusion model (HEM) where the jump sizes have a hyper-exponential distribution, i.e., a

mixture of a finite number of exponential distributions. For background on the HEM, see

2

Levendorskiı [30] and Cai and Kou [9]. The contribution of the current paper is three-fold:

(1) Even in the special case of the BSM, our approach is simpler as we essentially use only

Ito’s formula and do not need more advanced results such as those of Bessel processes and

Lamperti’s representation. See Section 3. (2) Our approach is more general as it applies to

the HEM (see Section 4). As a by-product we also show that under the HEM a well-known

recursion relating to Asian options has a unique solution in a probabilistic sense, and the

integral of the underlying asset price process at the exponential time has the same distribution

as a combination of a sequence of independent gamma and beta random variables; see Section

4.1. (3) The double-Laplace transform can be inverted numerically via a latest two-sided Euler

inversion algorithm along with a scaling factor proposed in Petrella [37].

We analyze the algorithm’s accuracy, stability, and low-volatility performance by conducting

a detailed comparison with other existing methods. For example our pricing method is highly

accurate compared with the benchmarks from the three existing pricing methods under the

BSM: (i) Linetsky’s method, (ii) Vecer’s method, and (iii) Geman and Yor’s single-Laplace

method via Shaw’s elegant Mathematica implementation. Moreover, our method performs well

even for low volatilities, e.g., 0.05. See Section 5.

The rest of the paper is organized as follows. Section 2 contains a general formulation of

the double-Laplace transform of Asian option prices. Section 3 concentrates on pricing Asian

options under the BSM. In Section 4, we extend the results in Section 3 to the more general

HEM. Section 5 is devoted to the implementation of our pricing algorithm via the latest two-

sided, two-dimensional Euler inversion algorithm with a scaling factor. Some proofs and some

numerical issues are presented in the appendices and the online supplement.

2 A Double-Laplace Transform

For simplicity, we shall focus on Asian call options, as Asian put options can be treated similarly.

The payoff of a continuous Asian call option with a mature time t and a fixed strike K is(

S0Att −K

)+, where At :=

∫ t0 eX(s)ds, S(t) is the underlying asset price process with S(0) ≡

3

S0, and X(t) := log(S(t)/S(0)) is the return process. Standard finance theory says that the

Asian option price at time zero can be expressed as P (t, k) := e−rtE(

S0Att −K

)+, where E

is expectation under a pricing probability measure P. Under the BSM the measure P is the

unique risk neutral measure, whereas under more general models P may be obtained in other

ways, such as using utility functions or mean variance hedging arguments. For more details,

see, e.g., Shreve [44].

A key component of our double-Laplace inversion method is a scaling factor X > S0. More

precisely, with k := ln( XKt) we can rewrite the option price P (t, k) = e−rtX

t E(

S0X At − e−k

)+.

Note that k can be either positive or negative, so the Laplace transform w.r.t. k will be two-

sided. The scaling factor X introduced by Petrella [37] is primarily to control the associated

discretization errors and to let the inversion occur at a reasonable point k; see also Cai et

al. [10], where they introduced a shift parameter for the two-sided Euler inversion algorithm.

Thanks to the scaling factor, the resulting inversion algorithm appears to be accurate, fast, and

stable even in the case of low volatility, e.g. σ = 0.05; see Section 5.

The following result presents an analytical representation for the double-Laplace transform

of f(t, k) := XE(S0X At− e−k)+ w.r.t. t and k. Note that Theorem 2.1 holds not only under the

BSM but also under other stochastic models. The result reduces the problem of pricing Asian

options to the study of real moments of the exponentially-stopped average E[Aν+1Tµ

].

Theorem 2.1. Let L(µ, ν) be the double-Laplace transform of f(t, k) w.r.t. t and k, respectively.

More precisely, L(µ, ν) =∫∞0

∫∞−∞ e−µte−νkXE(S0

X At − e−k)+dkdt. Then we have that

L(µ, ν) =XE[Aν+1

Tµ]

µν(ν + 1)

(S0

X

)ν+1

, µ > 0, ν > 0, (1)

where ATµ =∫ Tµ

0 eX(s)ds and Tµ is an exponential random variable with rate µ independent of

{X(t) : t ≥ 0}. Here µ > 0 and ν > 0 should satisfy E[Aν+1Tµ

] < +∞.

4

Proof : Applying Fubini’s theorem yields

L(µ, ν) = X

∫ ∞

0e−µtE

[∫ ∞

− ln(S0At/X)e−νk

(S0

XAt − e−k

)dk

]dt

= X

∫ ∞

0e−µtE

[S0

XAt

∫ ∞

− ln(S0At/X)e−νkdk −

∫ ∞

− ln(S0At/X)e−(ν+1)kdk

]dt

= X

∫ ∞

0e−µtE

[(S0At/X)ν+1

ν− (S0At/X)ν+1

ν + 1

]dt

= X

∫ ∞

0e−µt E[Aν+1

t ]ν(ν + 1)

(S0

X

)ν+1

dt =X

µν(ν + 1)

(S0

X

)ν+1

· E[Aν+1Tµ

],

from which the proof is completed. ¤

The idea of taking Laplace transform w.r.t. the log-strike ln(K) dates back to the work by

Carr and Madan [12]. Here we use the scaled log-strike ln(X/(Kt)) instead, as suggested in

Petrella [37]. A different double-Laplace transform was given in Fu et al. [21] under the BSM,

where the transform is taken w.r.t. t and K.

3 Pricing Asian Options under the BSM

In this section we study Asian option pricing under the BSM via the double-Laplace transforms.

More precisely, we investigate the distribution of ATµ so that we can compute E[Aν+1Tµ

] explicitly

and hence obtain analytical solutions for the double-Laplace transforms, thanks to Theorem

2.1.

3.1 Distribution of ATµ under the BSM

The classical BSM postulates that under the risk-neutral measure P, the return process {X(t) =

log(S(t)/S(0)) : t ≥ 0} is given by X(t) =(r − σ2

2

)t + σW (t), X(0) = 0, where r is the risk-

free rate, σ the volatility, and {W (t) : t ≥ 0} the standard Brownian motion. The infinitesimal

generator of {S(t)} is

Lf(s) =σ2

2s2f ′′(s) + rsf ′(s) (2)

5

for any twice continuously differentiable function f(·), and the Levy exponent of {X(t)} is

G(x) :=E

[exX(t)

]

t=

σ2

2x2 +

(r − σ2

2

)x. (3)

Let α1 and α2 be the two roots of the equation G(x) = µ(> 0) under the BSM. Then,

α1 =−v +

√v2 + 2µ

2> 0, α2 =

−v −√

v2 + 2µ

2< 0, (4)

where µ = 4µσ2 and v = 2r

σ2 − 1.

Consider the following nonhomogeneous ordinary differential equation (ODE)

Ly(s) = (s + µ)y(s)− µ, for s ≥ 0. (5)

Note that the equation (5) has two singularities, a regular singularity at 0 and an irregular

singularity at +∞. Due to the singularity, the above equation has infinitely many solutions.

However, if we impose an additional condition that the solution must be bounded, then the

solution is unique.

Theorem 3.1. (Uniqueness of the solution of the ODE (5) via a stochastic represen-

tation) A bounded solution to the ODE (5), if exists, must be unique. More precisely, suppose

a(s) solves the ODE (5) and sups∈[0,∞) |a(s)| ≤ C < ∞ for some constant C > 0. Then we

must have

a(s) = E[exp

(−sATµ

)]for any s ≥ 0. (6)

Proof: In terms of S(t), we can rewrite E[exp(−sATµ

)] as

E[exp(−sATµ

)] =Es

[∫ ∞

0µ exp

(−

∫ t

0[µ + S(u)]du

)dt

], (7)

where the notation Es means that the process {S(t)} starts from s, i.e. S(0) = s. By Ito’s

formula, we have that

Mt := a(S(t)) exp(−

∫ t

0[µ + S(u)]du

)+

∫ t

0µ exp

(−

∫ v

0[µ + S(u)]du

)dv

6

is a local martingale. Indeed, since

da(S(t)) = a′(S(t)) [rS(t)dt + σS(t)dW (t)] +12a′′(S(t))σ2S2(t)dt

= [(S(t) + µ)a(S(t))− µ] dt + a′(S(t))σS(t)dW (t),

where the last equality follows from the fact that a(s) solves the ODE (5), and

d exp(−

∫ t

0[µ + S(u)]du

)= exp

(−

∫ t

0[µ + S(u)]du

)· {−[µ + S(t)]} dt ,

we obtain by some algebra that

dMt = exp(−

∫ t

0[µ + S(u)]du

)· a′(S(t))σS(t)dW (t),

which implies that {Mt} is a local martingale. Actually, {Mt} is a true martingale as Mt is

uniformly bounded, supt≥0 |Mt| ≤ supt≥0

{C +

∫ t0 µe−µvdv

}= C + 1 < ∞, because µ > 0 and

S(u) ≥ 0. Thus, a(s) = a(S(0)) = Es[M0] = Es[Mt]. Letting t → +∞, the first term in Mt

goes to zero almost surely because a(·) is bounded, and therefore

Mt →∫ ∞

0µ exp

(−

∫ v

0{µ + S(u)}du

)dv,

almost surely. Accordingly, by the dominated convergence theorem,

a(s) = Es[ limt→∞Mt] = Es

[∫ ∞

0µ exp

(−

∫ v

0{µ + S(u)}du

)dv

]= E[exp

(−sATµ

)],

where the last equality holds due to (7). ¤

Theorem 3.1 implies that if we can find a particular bounded solution to the ODE (5),

it must have the stochastic representation in (6). To find such a one, consider a difference

equation (or a recursion) for a function H(ν) defined on (−1, α1)

h(ν)H(ν) = νH(ν − 1) for any ν ∈ (0, α1), and H(0) = 1, (8)

h(ν) ≡ µ−G(ν) = −σ2

2ν2 −

(r − σ2

2

)ν + µ = −σ2

2(ν − α1)(ν − α2). (9)

7

In general, if the above difference equation (8) has one solution H1(ν), then there exist an

infinite number of solutions to (8). In fact, any function in the following class

{H(ν) = θ(ν)H1(ν) : for some periodic function θ(ν) s.t. θ(ν) = θ(ν − 1) for any ν ∈ (0, α1)

}

also solves (8). This also partly explains why very few people investigated Asian option pricing

based on this recursion. However, we shall show next that the difference equation (8) has a

unique solution if we restrict our attention to random variables.

Theorem 3.2. (A particular bounded solution to the ODE (5)) If there exists a non-

negative random variable X such that H(ν) := E[Xν ] satisfies the difference equation (8), then

the Laplace transform of X, i.e. E[e−sX ], solves the nonhomogeneous ODE (5).

Proof: Denote the Laplace transform of X by y(s) = E[e−sX ], for s ≥ 0. Note that for any

a ∈ (0,min(α1, 1)), we have

∫ +∞

0s−ae−sXds = Γ(1− a)Xa−1 and

∫ +∞

0s−a−1

(e−sX − 1

)ds = −Γ(1− a)

aXa,

where the second equality holds due to integration by parts. Taking expectations on both sides

of the two equations above and applying Fubini’s theorem yields

E[Xa−1] =1

Γ(1− a)

∫ ∞

0s−ay(s)ds and E[Xa] = − a

Γ(1− a)

∫ ∞

0s−a−1 (y(s)− 1) ds.

Thus, by the difference equation (8), we have

− ah(a)Γ(1− a)

∫ ∞

0s−a−1 (y(s)− 1) ds =

a

Γ(1− a)

∫ ∞

0s−ay(s)ds,

i.e.

0 =∫ ∞

0s−a−1 [sy(s) + h(a)(y(s)− 1)] ds,

where h(a) is given by (9). Setting s = e−x and z(x) = y(s)− 1, we have

0 =∫ ∞

−∞eax

{e−x(z(x) + 1) + h(a)z(x)

}dx, for any a ∈ (0,min(α1, 1)).

8

For simplicity of notations, rewrite h(a) as h(a) = h0a2+h1a+h2, with h0 = −σ2

2 , h1 = −r+ σ2

2 ,

and h2 = µ. Note that integration by parts yields

∫ ∞

−∞eaxaz(x)dx = −

∫ ∞

−∞eaxz′(x)dx and

∫ ∞

−∞eaxa2z(x)dx =

∫ ∞

−∞eaxz′′(x)dx

because [z(x)eax]∣∣∣+∞

x=−∞= 0 and [z′(x)eax]

∣∣∣∞

x=−∞= 0. Then for any a ∈ (0,min(α1, 1)),

0 =∫ ∞

−∞eax

{e−x(z(x) + 1) +

(h0a

2 + h1a + h2

)z(x)

}dx

=∫ ∞

−∞eax

{e−x(z(x) + 1) + h0z

′′(x)− h1z′(x) + h2z(x)

}dx.

By the uniqueness of the moment generating function, we have an ODE

h0z′′(x)− h1z

′(x) + h2z(x) + e−x(z(x) + 1) = 0.

Now transferring the ODE for z(x) back to that for y(s), with s = e−x we have z(x) = y(s)−1,

z′(x) = −sy′(s), and z′′(x) = sy′(s) + s2y′′(s). Then the ODE becomes

h0s2y′′(s) + (h1 + h0)sy′(s) + (h2 + s)y(s) = h2.

Substituting h0, h1, and h2 into the above equation, we have the nonhomogeneous ODE (5). ¤

Theorem 3.3. Under the BSM, we have

ATµ=d2σ2

Z(1,−α2)Z(α1)

(10)

and therefore

E[AνTµ

] =(

2σ2

)ν Γ(ν + 1)Γ(α1 − ν)Γ(1− α2)Γ(α1)Γ(−α2 + ν + 1)

, for any ν ∈ (−1, α1). (11)

Here Z(a, b) denotes a beta random variable with parameters a and b, Z(a) a gamma random

variable with scale parameter 1 and shape parameter a, and Γ(·) the gamma function. Moreover,

Z(1,−α2) and Z(α1) are independent with α1 and α2 given by (4).

9

Proof: Consider a random variable χ such that χ =d2σ2 Z(1,−α2)/Z(α1). Then

E[χν ] =(

2σ2

)ν Γ(ν + 1)Γ(α1 − ν)Γ(1− α2)Γ(α1)Γ(−α2 + ν + 1)

, for any ν ∈ (−1, α1),

and furthermore, it can be easily verified that E[χν ] solves the difference equation (8). By

Theorem 3.2, we conclude that a∗(s) := E[e−sχ] for any s ≥ 0 is a particular bounded solution

to the ODE (5). As a result, it follows from Theorem 3.1 that ATµ =d χ =d2σ2 Z(1,−α2)/Z(α1),

which gives the distribution of ATµ . ¤

Remarks: 1. There are various ways to show that E[AνTµ

] satisfies recursions similar to (8)

for general processes. For example, Dufresne [20] used time reversal and Ito’s formula to derive

the recursion (8) for the BSM, and Carmona et al. [11] obtained similar recursions for general

Levy-type processes. In Appendix C we shall give a new proof for a recursion similar to (8)

under the HEM, although that proof is not needed to study Asian options.

2. The result (10) coincides with that in Yor [50] and Matsumoto and Yor [33]. However,

compared with the existing proofs involving Bessel processes and Lamperti’s representation,

our approach is simpler and more elementary. Furthermore, we illustrate in Section 4 that our

approach is more general, because it can be extended to the case of the HEM.

3.2 Pricing Formulae and Hedging Parameters under the BSM

Theorem 3.4. Under the BSM, for every µ and ν such that µ > 0 and ν ∈ (0, α1 − 1), the

double-Laplace transform of XE(S0X At − e−k)+ w.r.t. t and k is given by:

L(µ, ν) =X

µν(ν + 1)

(2S0

Xσ2

)ν+1 Γ(ν + 2)Γ(α1 − ν − 1)Γ(1− α2)Γ(α1)Γ(ν − α2 + 2)

. (12)

Therefore, the Asian option price is equal to:

P (t, k) =e−rt

tL−1 (L(µ, ν))

∣∣∣k=ln(X/Kt)

,

where L−1, a function of t and k, denotes the Laplace inversion of L. Furthermore, for any

maturity t and strike K, two common greeks delta ∆(P (t, k)) and gamma Γ(P (t, k)) can be

10

calculated as follows

∆(P (t, k)) =∂

∂S0P (t, k) =

e−rt

tL−1

(XSν

0

µν

(2

Xσ2

)ν+1 Γ(ν + 2)Γ(α1 − ν − 1)Γ(1− α2)Γ(α1)Γ(ν − α2 + 2)

)∣∣∣k=ln(X/Kt)

Γ(P (t, k)) =∂2

∂S20

P (t, k) =e−rt

tL−1

(XSν−1

0

µ

(2

Xσ2

)ν+1 Γ(ν + 2)Γ(α1 − ν − 1)Γ(1− α2)Γ(α1)Γ(ν − α2 + 2)

)∣∣∣k=ln(X/Kt)

.

Proof. Combining (11) with (1) yields (12). The two greeks can be obtained by interchanging

derivatives and integrals based on Theorem A. 12 on pp. 203-204 in Schiff [39]. ¤

Theorem 3.4 requires (0, α1 − 1) to be nonempty, i.e. α1 > 1, which means some Laplace

parameter µ > 0 may be disqualified. Nonetheless, a broad range of µ meets the requirement.

For example, it is sufficient to have µ > r, because G(1) = r, α1 solves the equation G(x) = µ

and G(x) − µ is a continuous function. Furthermore, this restriction µ > r does not present

any difficulty in term of numerical Laplace inversion.

4 Pricing Asian Options under the HEM

In the HEM, the asset return process {X(t) : t ≥ 0} under a risk-neutral measure P is given by

X(t) =(

r − 12σ2 − λζ

)t + σW (t) +

N(t)∑

i=1

Yi, X(0) = 0,

where r is the risk-free rate, σ the volatility, ζ := E(eY1) − 1 =∑m

i=1piηi

ηi−1 +∑n

j=1qjθj

θj+1 − 1,

{W (t) : t ≥ 0} the standard Brownian motion, {N(t) : t ≥ 0} a Poisson process with rate λ,

and {Yi : i ∈ N} i.i.d. hyper-exponentially distributed random variables with the probability

density function (pdf)

fY (y) =m∑

i=1

piηie−ηiyI{y≥0} +

n∑

j=1

qjθjeθjyI{y<0}, (13)

with pi > 0, ηi > 1, for i = 1, · · · ,m, qj > 0, ηj > 0, for j = 1, · · · , n, and∑m

i=1 pi+∑n

j=1 qj = 1.

Due to the jumps, the risk-neutral measure is not unique. Here we assume the risk-neutral

measure P is chosen within a rational expectations equilibrium setting such that the equilibrium

price of an option is given by the expectation under P of the discounted option payoff. For

11

details, refer to Kou [28]. It is worth mentioning that when m = n = 0 the HEM is reduced to

the BSM, and when m = n = 1 the HEM is reduced to the double exponential jump diffusion

model. The Levy exponent of {Xt} is given by

G(x) :=E

[exX(t)

]

t=

12σ2x2 +

(r − 1

2σ2 − λζ

)x + λ

m∑

i=1

piηi

ηi − x+

n∑

j=1

qjθj

θj + x− 1

(14)

for any x ∈ (−θ1, η1). It can be shown (see Cai [8]) that for any µ > 0, the equation G(x) = µ

has exactly (m + n + 2) real roots β1,µ, · · · , βm+1,µ, γ1,µ, · · · , γn+1,µ satisfying

−∞ < γn+1,µ < −θn < γn,µ < · · · < −θ1 < γ1,µ < 0 < β1,µ < η1 < · · · < βm,µ < ηm < βm+1,µ < ∞.(15)

Additionally, the infinitesimal generator of {S(t) = S(0)eX(t) : t ≥ 0} is given by

Lf(s) =σ2

2s2f ′′(s) + (r − λζ)sf ′(s) + λ

∫ +∞

−∞[f(seu)− f(s)]fY (u)du, (16)

for any twice continuously differentiable function f(·).

4.1 Distribution of ATµ under the HEM

Consider the following nonhomogeneous ordinary integro-differential equation (OIDE)

Ly(s) = (s + µ)y(s)− µ, (17)

where L is given by (16). Similarly as in the BSM, we have the following theorem.

Theorem 4.1. (Uniqueness of the solution of the OIDE (17) via a stochastic repre-

sentation) There is at most one bounded solution to the OIDE (17). More precisely, suppose

a(s) solves (17) and sups∈[0,∞) |a(s)| ≤ C < ∞ for some constant C > 0. Then we must have

a(s) = E[exp

(−sATµ

)]for any s ≥ 0. (18)

Proof : See Appendix A. ¤

Since the proofs for Theorem 3.1 and 4.1 involve only Ito’s formula (see Section 1.2 and

1.3 of Øksendal and Sulem [36] or Applebaum [5] for more general Ito formulae), the result

12

holds for more general underlying process S(t) such as exponential Levy processes and Levy

diffusions. Next, we look for a particular bounded solution to the OIDE (17), which has the

stochastic representation in (18). Consider a difference equation (or a recursion) for a function

H(ν) defined on (−1, β1)

h(ν)H(ν) = νH(ν − 1) for any ν ∈ (0, β1), and H(0) = 1, (19)

where

h(ν) ≡µ−G(ν) = µ− 12σ2ν2 − (r − 1

2σ2 − λζ)ν − λ

m∑

i=1

piηi

ηi − ν+

n∑

j=1

qjθj

θj + ν− 1

=(

σ2

2

) ∏m+1i=1 (βi − ν)

∏n+1j=1 (−γj + ν)∏m

i=1(ηi − ν)∏n

j=1(θj + ν). (20)

Here β1, · · · , βm+1, γ1, · · · , γn+1 are actually β1,µ, · · · , βm+1,µ, γ1,µ, · · · , γn+1,µ, which are

(m + n + 2) real roots of the equation G(x) = µ satisfying the condition (15).

Theorem 4.2. (A particular bounded solution to the OIDE (17)) If there is a nonneg-

ative random variable X such that H(ν) := E[Xν ] satisfies the difference equation (19), then

the Laplace transform of X, i.e. E[e−sX ], solves the nonhomogeneous OIDE (17).

Proof : See Appendix B. ¤

Theorem 4.3. Under the HEM, we have

ATµ=d2σ2

Z(1,−γ1)∏n

j=1 Z(θj + 1,−γj+1 − θj)Z(βm+1)

∏mi=1 Z(βi, ηi − βi)

, (21)

where all the gamma and beta random variables on the RHS are independent, and therefore for

any ν ∈ (−1, β1),

E[AνTµ

]

=(

2σ2

)ν Γ(1 + ν)Γ(1− γ1)Γ(1− γ1 + ν)

·n∏

j=1

[Γ(θj + 1 + ν)Γ(1− γj+1)Γ(1− γj+1 + ν)Γ(θj + 1)

]·

m∏

i=1

[Γ(βi − ν)Γ(ηi)Γ(ηi − ν)Γ(βi)

]· Γ(βm+1 − ν)

Γ(βm+1).

(22)

13

Proof: Consider a random variable χ that is defined by the right side of (21). Then simple

algebra shows that E[χν ] is given by the right side of (22). Moreover, we can verify that E[χν ]

solves the difference equation (19). By Theorem 4.2, a∗(s) := Ee−sχ, for any s ≥ 0, is a

particular bounded solution to the OIDE (17), and the proof is terminated via Theorem 4.1. ¤

4.2 Pricing Formulae for Asian Options and Hedging Parameters under theHEM

Theorem 4.4. Under the HEM, for every µ and ν such that µ > 0 and ν ∈ (0, β1 − 1), the

double-Laplace transform of XE(S0X At − e−k)+ w.r.t. t and k is given by:

L(µ, ν) =X

µν(ν + 1)

(2S0

Xσ2

)ν+1 Γ(2 + ν)Γ(1− γ1)Γ(2− γ1 + ν)

·n∏

j=1

[Γ(θj + 2 + ν)Γ(1− γj+1)Γ(2− γj+1 + ν)Γ(θj + 1)

]·

m∏

i=1

[Γ(βi − ν − 1)Γ(ηi)Γ(ηi − ν − 1)Γ(βi)

]· Γ(βm+1 − ν − 1)

Γ(βm+1). (23)

Therefore, the Asian option price is equal to

P (t, k) =e−rt

tL−1 (L(µ, ν)) |k=ln(X/Kt).

And two common greeks delta ∆(P (t, k)) and gamma Γ(P (t, k)) can be calculated as follows

∆(P (t, k)) =∂

∂S0P (t, k) =

e−rt

tL−1

(XSν0

µν(

2Xσ2

)ν+1 · Γ(2 + ν)Γ(1− γ1)Γ(2− γ1 + ν)

·n∏

j=1

[Γ(θj + 2 + ν)Γ(1− γj+1)Γ(2− γj+1 + ν)Γ(θj + 1)

]·

m∏

i=1

[Γ(βi − ν − 1)Γ(ηi)Γ(ηi − ν − 1)Γ(βi)

]· Γ(βm+1 − ν − 1)

Γ(βm+1)

)∣∣∣k=ln(X/Kt)

Γ(P (t, k)) =∂2

∂S20

P (t, k) =e−rt

tL−1

(XSν−10

µ(

2Xσ2

)ν+1 · Γ(2 + ν)Γ(1− γ1)Γ(2− γ1 + ν)

·n∏

j=1

[Γ(θj + 2 + ν)Γ(1− γj+1)Γ(2− γj+1 + ν)Γ(θj + 1)

]·

m∏

i=1

[Γ(βi − ν − 1)Γ(ηi)Γ(ηi − ν − 1)Γ(βi)

]· Γ(βm+1 − ν − 1)

Γ(βm+1)

)∣∣∣k=ln(X/Kt)

Proof. The proof is similar to that of Theorem 3.4. ¤

5 Pricing Asian Options via a Two-Sided Euler Inversion Al-gorithm with a Scaling Factor

In this section, we intend to price Asian options under both the BSM and the HEM by inverting

L(µ, ν) in (12) and (23) numerically. The algorithm used here is proposed in Petrella [37],

14

which, as a generalization of the one-sided Euler inversion algorithm ([1] and [14]), introduces

a two-sided Euler inversion with a scaling factor.

The inversion formula in Petrella [37] to get f(t, k) from its Laplace transform L(µ, ν) is

f(t, k) =exp(A1/2 + A2/2)

4tk

×{L

(A1

2t,A2

2k

)+ 2

∞∑

s=0

(−1)sRe

[−L

(A1

2t,A2

2k− iπ

k− isπ

k

)]

+ 2∞∑

s=0

(−1)sRe

[−L

(A1

2t− iπ

t− isπ

t,A2

2k

)]

+ 2∞∑

j=0

(−1)jRe

[ ∞∑

s=0

(−1)sL(

A1

2t− iπ

t− ijπ

t,A2

2k− iπ

k− isπ

k

)]

+ 2∞∑

j=0

(−1)jRe

[ ∞∑

s=0

(−1)sL(

A1

2t− iπ

t− ijπ

t,A2

2k+

iπ

k+

isπ

k

)] }

− e+d − e−d , (24)

where the two errors are given by

e+d =

∞∑

j2=1

∞∑

j1=0

e−(j1A1+j2A2)f((2j1 + 1)t, (2j2 + 1)k) +∞∑

j1=1

e−j1A1f((2j1 + 1)t, k), (25)

e−d =−1∑

j2=−∞

∞∑

j1=0

e−(j1A1+j2A2)f((2j1 + 1)t, (2j2 + 1)k), (26)

and A1 and A2 (to be specified later) are some inversion parameters used to control the errors.

The inversion appears to be accurate, stable, and easy to implement. For example, under

the BSM, the prices produced via our algorithm highly agree with benchmarks generated by the

other three important methods by Linetsky, Geman-Yor-Shaw, and Vecer, and it is stable even

for low volatilities (e.g. σ = 0.05). The algorithm is easy to implement primarily because the

closed-form Laplace transform involves only gamma functions. We also study the algorithm’s

stability, derive a discretization error bound, and conduct a comparison with the Fourier and

Laplace inversion algorithm by Fusai [22]; see the online supplement.

In the inversion, we have to calculate alternating series of the form∑∞

i=0(−1)iai in the

expression (24). To accelerate the convergence rate, we adopt the idea of Euler transformation

15

(see [1] and [14]), and approximate∑∞

i=0(−1)iai by E(m,n) :=∑m

k=0m!

k!(m−k)!2−mSn+k, where

Sj :=∑j

i=0(−1)iai. Since there are two transforms, Euler transformation will be used twice.

We use (n1,m1) and (n2,m2) to express parameters involved in Euler transformations for Euler

inversion w.r.t. t and k, respectively. More precisely, for the first infinite sum and the inner

series of both the third and fourth double sum on the RHS of (24), we use Euler transformation

with parameters (n2,m2); while for the second infinite sum and the outer series of both the

third and fourth double sum on the RHS of (24), we use Euler transformation with parameters

(n1,m1). As suggested in Abate and Whitt [1], we shall set m1 = n1 + 15 and m2 = n2 + 15.

In the inversion algorithm there are several parameters to be chosen: (1) (n1, n2). The larger

n1 and n2 will lead to better accuracy at the cost of computation time. In our experiments,

n1 = n2 = 35 seems to achieve excellent accuracy even for low volatilities (e.g. σ = 0.05);

usually even n1 = 15 and n2 = 35 can yield good accuracy if the volatility is not low. (2)

(A1, A2). (3) The scaling factor X. One appealing feature of the inversion algorithm is its

insensitivity to the choices of A1, A2, and X even for low volatilities. In other words, for

wide ranges of parameters A1, A2, and X, the numerical results are almost identical. This

is illustrated in Figure 1 in the online supplement. For convenience, in most cases we simply

select A1 = 28, A2 = 40, and, as suggested in Petrella [37],

X = Kt · ek = Kt · exp{

min(

A2

2/(σ√

t),

A2

4

)}. (27)

for practical implementation, although one can freely choose other values.

5.1 Pricing Asian Options under the BSM

5.1.1 Comparison of Accuracy with Other Methods

To check the accuracy of our double-Laplace (DL) inversion algorithm, we consider seven test

cases in Table 1, which are frequently used in the literature ([21], [32], [43], [17], [48]). A

detailed comparison of accuracy between our double-Laplace inversion algorithm with those

obtained from six other existing methods is given in Table 2. From Table 2, we can see that our

DL prices highly agree with benchmarks of existing methods. Specifically, our DL prices agree

16

Case No. S0 K r σ t1 2.0 2 0.02 0.10 12 2.0 2 0.18 0.30 13 2.0 2 0.0125 0.25 24 1.9 2 0.05 0.50 15 2.0 2 0.05 0.50 16 2.1 2 0.05 0.50 17 2.0 2 0.05 0.50 2

Table 1: Seven test cases in the literature.

with both Geman-Yor-Shaw’s and Linetsky’s results to ten decimal points and with Vecer’s

results to six decimal points. Moreover, our DL prices also agree with Zhang’s PDE results to

six decimal points. These consistence indicates that our double-Laplace inversion algorithm is

very accurate and can also be used as benchmarks for pricing Asian options.

Case DL Prices Linetsky GY-Shaw Vecer Zhang GYS-Mellin MAE31 0.0559860415 0.0559860415 0.0559860415 0.055986 0.055986 0.0559856 0.0559862 0.2183875466 0.2183875466 0.2183875466 0.218388 0.218388 0.218359 0.2186633 0.1722687410 0.1722687410 0.1722687410 0.172269 0.172269 0.172263 0.1722634 0.1931737903 0.1931737903 0.1931737903 0.193174 0.193174 0.193060 0.1931885 0.2464156905 0.2464156905 0.2464156905 0.246416 0.246416 0.246522 0.2463826 0.3062203648 0.3062203648 0.3062203648 0.306220 0.306220 0.306501 0.3061397 0.3500952190 0.3500952190 0.3500952190 0.350095 0.350095 0.348924 0.349909

Table 2: Comparison of accuracy with other existing methods. The parameters associated with our double-Laplace (DL) inversion method are n1 = 35, n2 = 55, A1 = 28, A2 = 40, and X given by (27). Results ofLinetsky’s eigenfunction expansion method are taken from Table 3 in [32]. Vecer’s PDE results are from TableA in of [48]. The “ GYS-Mellin” numbers are taken from a Mellin transformed-based approximation in [43]. Theother three columns, including “GY-Shaw,” “Zhang,” and “MAE3”, are all taken from the table on p. 383 in[17], and correspond to the methods in [41], [51], and [17], respectively. Our numerical DL prices are calculatedusing Matlab 7.1 on a desktop with Quad CPU 2.66 GHz.

Although Shaw’s GYS-Mellin results and Dewynne and Shaw’s MAE3 results seem less

accurate than other methods, they are still sufficiently accurate in practice and moreover, these

two methods have their own advantages. First, Shaw’s GYS-Mellin method turns out to be very

fast. For example, in Case 1 in Table 2 and on a desktop with Quad CPU 2.66 GHz, it takes

only 0.047 seconds to produce one result; whereas our pricing method requires 0.563 seconds

to match the GYS-Mellin result to five decimal points. Second, these two methods work better

than most other methods when the volatility is extremely low; for details, see Section 5.1.2.

Note that although in our theorems the dividend δ = 0, they can be easily extended to the

case of nonzero dividends. Indeed for the two families of the extended seven cases in Section 6.2

17

(δ > r) and Section 6.3 (δ = r) in [17], our DL prices, GY-Shaw prices (or its variant CIBess

prices) and Zhang’s results agree with one another to six or seven decimal points, hence being

more accurate than MAE3; see Table 3.

Extension of Seven Cases in Table 1 as δ > r (see Section 6.2 in [17]).Case S0 K r δ σ t DL Prices GY-Shaw MAE3 Zhang1∗ 2.0 2 0.02 0.04 0.1 1 0.0357853 0.0357853 0.0357854 0.03578532∗ 2.0 2 0.18 0.36 0.3 1 0.0522607 0.0522607 0.0522755 0.05226073∗ 2.0 2 0.0125 0.025 0.25 2 0.145308 0.145308 0.145310 0.1453084∗ 1.9 2 0.05 0.1 0.5 1 0.147562 0.147562 0.147618 0.1475625∗ 2.0 2 0.05 0.1 0.5 1 0.191747 0.191747 0.191760 0.1917476∗ 2.1 2 0.05 0.1 0.5 1 0.242316 0.242316 0.242283 0.2423167∗ 2.0 2 0.05 0.1 0.5 2 0.240495 0.240495 0.240564 0.240495

Extension of Seven Cases in Table 1 as δ = r (see Section 6.3 in [17]).Case S0 K r δ σ t DL Prices CIBess MAE3 Zhang1∗∗ 2.0 2 0.02 0.02 0.1 1 0.0451431 0.0451431 0.0451431 0.04514312∗∗ 2.0 2 0.18 0.18 0.3 1 0.115188 0.115188 0.115188 0.1151883∗∗ 2.0 2 0.0125 0.0125 0.25 2 0.158380 0.158380 0.158378 0.1583804∗∗ 1.9 2 0.05 0.05 0.5 1 0.169202 0.169202 0.169238 0.1692025∗∗ 2.0 2 0.05 0.05 0.5 1 0.217815 0.217815 0.217805 0.2178156∗∗ 2.1 2 0.05 0.05 0.5 1 0.272924 0.272924 0.272869 0.2729247∗∗ 2.0 2 0.05 0.05 0.5 2 0.291315 0.291315 0.291264 0.291315

Table 3: Comparison of accuracy in extended cases when δ > r or δ = r

5.1.2 Comparison of Behaviors for Low Volatilities

It is well known that many numerical methods for Asian option pricing do not perform well for

low volatilities (see [15, 21, 42]). Here we would like to conduct cross-comparisons of behaviors

for reasonably low (e.g., σ = 0.05) and extremely low volatilities (e.g., σ ≤ 0.01) between our

double-Laplace inversion method and the three methods discussed in [17], GY-Shaw (or its

variants GYS-Full and CIBess), MAE3, and Zhang’s method, for three cases δ < r, δ > r and

δ = r, where δ denotes the dividend (in [17] the dividend is denoted by q).

To investigate the behaviors of these algorithms in the case of low volatilities, we modify

the test case 1 in Table 1 and the test cases 1∗ and 1∗∗ in Table 3 by letting the volatility σ

be small; see Table 4. Note that the parameter settings are the same as in Section 6.1–6.3 in

[17]. Table 4 demonstrates that when the volatility is reasonably small (= 0.05), the numerical

results obtained by the four methods coincide with one another to six or seven decimal points.

18

Nonetheless, when the volatility is extremely small (≤ 0.01), MAE3 and Zhang’s results are

still available and highly agree with each other, but our double-Laplace inversion method and

GY-Shaw methods no longer work. Consequently, we conclude that our methods and GY-Shaw

methods are reliable in the case of normal or reasonably low volatilities (σ ≥ 0.05). However, we

may use Dewynne and Shaw’s asymptotic method or Zhang’s PDE method for the extremely

low volatilities (σ < 0.05).

Extension of Case 1 in Table 1 when δ < r and σ is extremely small (Section 6.1 in [17]).Case S0 K r δ σ t DL Prices GY-Shaw MAE3 Zhang

1 2.0 2 0.02 0 0.1 1 0.0559860 0.0559860 0.0559860 0.05598601A 2.0 2 0.02 0 0.05 1 0.0339412 0.0339412 0.0339412 0.03394121B 2.0 2 0.02 0 0.01 1 NA NA 0.0199278 0.01992781C 2.0 2 0.02 0 0.005 1 NA NA 0.0197357 0.01973571D 2.0 2 0.02 0 0.001 1 NA NA 0.0197353 0.0197353Extension of Case 1∗ in Table 3 when δ > r and σ is extremely small (see Section 6.2 in [17]).

Case S0 K r δ σ t DL Prices GYS-full MAE3 Zhang1 ∗ 2.0 2 0.02 0.04 0.1 1 0.0357853 0.0357853 0.0357854 0.03578531A∗ 2.0 2 0.02 0.04 0.05 1 0.0140247 0.0140247 0.0140248 0.01402471B∗ 2.0 2 0.02 0.04 0.01 1 NA NA 0.000190254 0.0001902541C∗ 2.0 2 0.02 0.04 0.005 1 NA NA 3.7991×10−7 3.7993×10−7

1D∗ 2.0 2 0.02 0.04 0.001 1 NA NA o(10−70) o(10−72)Extension of Case 1∗∗ in Table 3 when δ = r and σ is extremely small (see Section 6.3 in [17]).

Case S0 K r δ σ t DL Prices CIBess MAE3 Zhang1∗∗ 2.0 2 0.02 0.02 0.1 1 0.0451431 0.0451431 0.0451431 0.0451431

1A∗∗ 2.0 2 0.02 0.02 0.05 1 0.0225755 0.0225755 0.0225755 0.02257551B∗∗ 2.0 2 0.02 0.02 0.01 1 NA NA 0.00451536 0.004515361C∗∗ 2.0 2 0.02 0.02 0.005 1 NA NA 0.00225768 0.002257681D∗∗ 2.0 2 0.02 0.02 0.001 1 NA NA 0.000451537 0.000451537

Table 4: Comparison of accuracy for extremely low volatilities when δ < r, δ > r or δ = r

5.2 Pricing Asian Options under the HEM

In this section, we price Asian options numerically under the HEM by inverting the double-

Laplace transform (23) via the two-sided Euler inversion algorithm. Without loss of generality,

we concentrate on Kou’s model, which along with the BSM are the most important special cases

of the HEM. Table 5 and Table 6 provide the numerical results of Asian option prices using

double-Laplace transform in the cases of λ = 3 and λ = 5, respectively. The double-Laplace

inversion method seems to be still accurate under Kou’s model. Furthermore, our algorithm is

19

efficient because it takes only around 6 seconds to produce one numerical result on a desktop

with Quad CPU 2.66 GHz.

σ K DL Prices MC Prices Std Err Abs Err Rel Err0.05 90 13.41924 13.42054 0.00048 -0.00130 0.0097%0.05 95 8.98812 8.98730 0.00095 0.00082 0.0091%0.05 100 4.95673 4.95681 0.00162 -0.00008 0.0016%0.05 105 2.13611 2.13453 0.00208 0.00158 0.0740%0.05 110 0.83091 0.82995 0.00193 0.00096 0.1200%0.1 90 13.48451 13.47574 0.00071 0.00877 0.0651%0.1 95 9.20478 9.20559 0.00135 -0.00081 0.0088%0.1 100 5.53662 5.53619 0.00207 0.00043 0.0078%0.1 105 2.88896 2.88890 0.00249 0.00006 0.0021%0.1 110 1.33809 1.33781 0.00238 0.00028 0.0210%0.2 90 14.03280 14.03489 0.00193 -0.00289 0.0206%0.2 95 10.32293 10.32461 0.00276 -0.00168 0.0163%0.2 100 7.21244 7.21556 0.00343 -0.00312 0.0432%0.2 105 4.78516 4.78822 0.00380 -0.00306 0.0638%0.2 110 3.02270 3.02558 0.00380 -0.00288 0.0952%0.3 90 15.19639 15.19689 0.00350 -0.00050 0.0033%0.3 95 11.92926 11.93168 0.00431 -0.00242 0.0203%0.3 100 9.14769 9.15063 0.00495 -0.00294 0.0321%0.3 105 6.86049 6.86412 0.00533 -0.00363 0.0529%0.3 110 5.04029 5.04400 0.00545 -0.00331 0.0656%0.4 90 16.68984 16.69294 0.00506 -0.00310 0.0186%0.4 95 13.73384 13.73747 0.00586 -0.00363 0.0264%0.4 100 11.17115 11.17579 0.00649 -0.00464 0.0415%0.4 105 8.99114 8.99645 0.00692 -0.00531 0.0590%0.4 110 7.16816 7.17317 0.00716 -0.00501 0.0698%0.5 90 18.35379 18.35851 0.00658 -0.00472 0.0257%0.5 95 15.62810 15.63389 0.00738 -0.00579 0.0370%0.5 100 13.22860 13.23485 0.00804 -0.00625 0.0472%0.5 105 11.13944 11.14602 0.00853 -0.00658 0.0590%0.5 110 9.33799 9.34396 0.00887 -0.00597 0.0639%

Table 5: Numerical results of Asian option prices under Kou’s model with λ = 3. Other parameters of themodel are set as: S0 = 100, r = 0.09, t = 1.0, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25. Parameters of thealgorithm are set as: n1 = 35, n2 = 55, A1 = 38.9, A2 = 40, and X = Kt exp{min(A2/θ, A2/10)}, whereθ = 2/

p(σ2 + 2p1λ/η2

1 + 2q1λ/θ21)t; DL prices are obtained by double-Laplace inversion; MC price are Monte

Carlo simulation estimates obtained by simulating 1 million paths and setting the discretization step size to be0.0001; Std Err is the standard error of the MC price; Abs Err and Rel Err are absolute and relative errors,respectively.

Appendix A. Proof of Theorem 4.1

Proof: For the HEM, the argument is similar to that for the BSM in Theorem 3.1 except that we

need to show the process {M(t)} is still a local martingale in the jump diffusion case. Indeed,

20

σ K LL Price MC Price Std Err Error0.05 90 13.47952 13.47729 0.00075 0.002230.05 95 9.16588 9.16468 0.00135 0.001200.05 100 5.38761 5.38519 0.00208 0.002420.05 105 2.72681 2.72275 0.00252 0.004060.05 110 1.28264 1.28034 0.00242 0.002300.1 90 13.55964 13.56384 0.00102 -0.004200.1 95 9.41962 9.42350 0.00173 -0.003880.1 100 5.91537 5.91707 0.00246 -0.001700.1 105 3.35071 3.35124 0.00287 -0.000530.1 110 1.74896 1.74934 0.00281 -0.000380.2 90 14.17380 14.17586 0.00217 -0.002060.2 95 10.53795 10.53973 0.00300 -0.001780.2 100 7.48805 7.48864 0.00367 -0.000590.2 105 5.09001 5.09000 0.00405 0.000010.2 110 3.32061 3.31967 0.00409 0.000960.3 90 15.33688 15.33728 0.00367 -0.000400.3 95 12.10723 12.10732 0.00448 -0.000090.3 100 9.35336 9.35297 0.00511 0.000390.3 105 7.08059 7.07908 0.00551 0.001510.3 110 5.26109 5.25875 0.00565 0.002340.4 90 16.81490 16.81475 0.00518 0.000150.4 95 13.87995 13.87950 0.00598 0.000450.4 100 11.33257 11.33142 0.00662 0.001150.4 105 9.16131 9.15913 0.00706 0.002180.4 110 7.34063 7.33721 0.00730 0.003420.5 90 18.46259 18.46136 0.00667 0.001230.5 95 15.75006 15.74865 0.00748 0.001410.5 100 13.36027 13.35842 0.00814 0.001850.5 105 11.27716 11.27410 0.00864 0.003060.5 110 9.47826 9.47384 0.00898 0.00443

Table 6: Numerical results of Asian option prices under Kou’s model with λ = 5. Other parameters ofthe model are set as: S0 = 100, r = 0.09, t = 1.0, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25. Parametersof the algorithms are n1 = 35, n2 = 55, A1 = 35.9, A2 = 40, and X = Kt exp{min(A2/θ, A2/10)}, whereθ = 2/

p(σ2 + 2p1λ/η2

1 + 2q1λ/θ21)t. DL prices are obtained by the double Laplace inversion; MC prices are

Monte Carlo simulation estimates by using 1 million paths and setting the discretization step size to be 0.0001.

by Ito’s formula for jump diffusions, we have

da(S(t)) = a′(S(t−))dSc(t) +12a′′(S(t−))d〈Sc, Sc〉(t) + d

∑

0<u≤t

[a(S(u))− a(S(u−))]

=[(r − λζ)S(t−)a′(S(t−)) +

12σ2S2(t−)a′′(S(t−))

]dt + σS(t−)a′(S(t−))dW (t)

+d∑

0<u≤t

[a(S(u))− a(S(u−))].

21

Since a(s) solves the OIDE (17), we have

da(S(t)) = [(S(t−) + µ)a(S(t−))− µ]dt + σS(t−)a′(S(t−))dW (t)

+d∑

0<u≤t

[a(S(u))− a(S(u−))]− λ

∫ +∞

−∞[a(S(t−)ey)− a(S(t−))] fY (y)dydt.

Note that

dM(t) = exp(−

∫ t

0{µ + S(u)}du

)· da(S(t)) + a(S(t−)) · d exp

(−

∫ t

0{µ + S(u)}du

)

+d⟨a(S(t), exp

(−

∫ t

0{µ + S(u)}du

) ⟩+ µ exp

(−

∫ t

0{µ + S(u)}du

)dt.

Plugging d(a(S(t)) into dM(t) yields

dM(t) = exp(−

∫ t

0{µ + S(u)}du

)a′(S(t−))σS(t−)dW (t)

+ exp(−

∫ t

0{µ + S(u)}du

)d

∑

0<u≤t

[a(S(u))− a(S(u−))]

−λ exp(−

∫ t

0{µ + S(u)}du

) ∫ +∞

−∞[a(S(t−)ey)− a(S(t−))] fY (y)dydt,

which is a local martingale. Then the same proof as in Theorem 3.1 applies. ¤

Appendix B. Proof of Theorem 4.2

Proof: Similar algebra as in Theorem 3.2 yields that

0 =∫ ∞

−∞eax

{e−x(z(x) + 1) + h(a)z(x)

}dx, for any a ∈ (0,min(β1, 1)), (28)

where β1 is the smallest positive root of G(x) = µ and z(x) = y(s)− 1. Plugging h(a) in (20)

into (28), we have that for all a ∈ (0,min(β1, 1)),∫ ∞

−∞eax

{e−x(z(x) + 1) +

[−1

2σ2a2 −

(r − σ2

2− λζ

)a + µ

]z(x)

}dx

+∫ ∞

−∞eax

{− λ

m∑

i=1

piηi

ηi − a+

n∑

j=1

qjθj

θj + a− 1

z(x)

}dx = 0.

Using the same technique as in the proof of Theorem 3.2, we can show that the first integral

on the left hand side of the above is equal to∫ ∞

−∞eax

{e−x(z(x) + 1)− 1

2σ2z′′(x) +

(r − σ2

2− λζ

)z′(x) + µz(x)

}dx.

22

In addition, we claim that the second integral is equal to∫ ∞

−∞eax

{−λ

∫ ∞

−∞[z(x− u)− z(x)] fY (u)du

}dx.

Indeed, this is because∫ ∞

−∞eax

{−λ

∫ ∞

−∞[z(x− u)− z(x)] fY (u)du

}dx

= −λ

∫ ∞

−∞fY (u)

{∫ ∞

−∞eaxz(x− u)dx

}du +

∫ ∞

−∞eaxλz(x)dx

= −λ

∫ ∞

−∞fY (u)

{∫ ∞

−∞ea(u+π)z(π)dπ

}du +

∫ ∞

−∞eaxλz(x)dx

=∫ ∞

−∞eaπz(π)

{−λ

∫ ∞

−∞eaufY (u)du

}dπ +

∫ ∞

−∞eaxλz(x)dx

=∫ ∞

−∞eax

{− λ

m∑

i=1

piηi

ηi − a+

n∑

j=1

qjθj

θj + a− 1

z(x)

}dx,

where the second equality is via change of variable x − u = π, and the last equality holds

because 0 < a < β1 < η1.

Thus we have∫ ∞

−∞eax

{e−x(z(x) + 1)− 1

2σ2z′′(x) +

(r − σ2

2− λζ

)z′(x) + µz(x)

− λ

∫ ∞

−∞[z(x− u)− z(x)] fY (u)du

}dx = 0, for all a ∈ (0,min(β1, 1)).

By the uniqueness of the moment generating function we have an OIDE as follows

e−x(z(x)+1)− 12σ2z′′(x)+

(r − σ2

2− λζ

)z′(x)+µz(x)−λ

∫ ∞

−∞[z(x− u)− z(x)] fY (u)du = 0.

Now transferring the OIDE back to y(s), with s = e−x we have

z(x) = y(s)− 1, z′(x) = −sy′(s), z′′(x) = sy′(s) + s2y′′(s),

z(x− u) = z(− log s− u) = z(− log(seu)) = y(seu)− 1

and the OIDE becomes

−σ2

2s2y′′(s)− (r − λζ) sy′(s) + (s + µ)y(s)− λ

∫ ∞

−∞[y(seu)− y(s)] fY (u)du = µ,

i.e., Ly(s) = (s + µ)y(s)− µ, from which the proof is completed. ¤

23

Appendix C. On the Recursion (19) under the HEM

In this section, we shall give an alternative proof that under the HEM, E[AνTµ

] solves the

recursion (19) using the Feynman-Kac formula. More precisely, under the HEM, E[AνTµ

] satisfies

the recursion (19), i.e.,

h(ν)E[AνTµ

] = νE[Aν−1Tµ

] for any ν ∈ (0, β1), (29)

where h(ν) is given by (20).

Proof: Under the HEM, define Y (t) =∫ t0 S(u)du, where S(u) represents the underlying stock

price at the time u. Consider a function f∗(t, x, y) defined as follows:

f∗(t, x, y) := E (Y νT | S(t) = x, Y (t) = y) , t ∈ [0, T ], x ∈ R+, y ∈ R,

where T is any fixed positive real number and ν ∈ (−1, β1) is a constant. Note that {(S(t), Y (t)) :

t ≥ 0} is a two-dimensional Markov process under the HEM so that f∗(t, S(t), Y (t)) =

E (Y νT | Ft). Accordingly, the multivariate Feynman-Kac theorem implies that f∗(t, x, y) solves

the following PIDE:

f∗t (t, x, y) + (r − λζ)xf∗x(t, x, y) + xf∗y (t, x, y) +σ2

2x2f∗xx(t, x, y)

+ λ

∫ +∞

−∞[f∗(t, xez, y)− f∗(t, x, y)]f∗Y (z)dz = 0, t ∈ [0, T ], x ∈ R+, y ∈ R.

On the other hand, by the Markovian property, we can rewrite f∗(t, x, y) as follows.

f∗(t, x, y) = E

[(y + x

∫ T−t

0e(r−λζ−σ2

2)u+σW (u)+

PN(u)i=1 Yidu

)ν]

= E [(y + xAT−t)ν ] .

Introduce a new function

f∗(t, x, y) := f∗(T − t, x, y) = E [(y + xAt)ν ] , t ∈ [0, T ], x ∈ R+, y ∈ R,

which then satisfies the following PIDE

−f∗t (t, x, y) + (r − λζ)xf∗x(t, x, y) + xf∗y (t, x, y) +σ2

2x2f∗xx(t, x, y)

+λ

∫ +∞

−∞[f∗(t, xez, y)− f∗(t, x, y)]f∗Y (z)dz = 0, t ∈ [0, T ], x ∈ R+, y ∈ R, (30)

24

with the initial value

f∗(0, x, y) = yν .

It is worth noting that the “T”in the PIDE (30) can be any positive real number.

Next, for any ν ∈ (0, β1), define h∗(x, y) = E[(

y + xATµ

)ν] =∫∞0 µe−µtf∗(t, x, y)dt, x ∈

R+, y ∈ R. Taking the Laplace transform w.r.t. t on both sides of the PIDE (30) and then

multiplying both sides by µ, we can show that h∗(x, y) solves a PIDE as follows.

µyν − µh∗(x, y)+(r − λζ)xh∗x(x, y) + xh∗y(x, y) +σ2

2x2h∗xx(x, y)

+λ

∫ +∞

−∞[h∗(xez, y)− h∗(x, y)]f∗Y (z)dz = 0, x ∈ R+, y ∈ R. (31)

Interchanging the derivatives and integrals by using Theorem A. 12 on the pp. 203-204 in Schiff

[39], we have

xh∗x(x, y) = νE[(

y + xATµ

)ν]− νyE[(

y + xATµ

)ν−1],

xh∗y(x, y) = νxE[(

y + xATµ

)ν−1],

x2h∗xx(x, y) = ν(ν − 1)E[(

y + xATµ

)ν] + ν(ν − 1)y2E[(

y + xATµ

)ν−2]

−ν(ν − 1)2yE[(

y + xATµ

)ν−1].

Substituting above into (31) yields

µyν +(−µ + (r − λζ)ν +

σ2

2ν(ν − 1)

)E

[(y + xATµ

)ν]

+ ν(x− (r − λζ)y − σ2(ν − 1)y

)E

[(y + xATµ

)ν−1]

+σ2

2ν(ν − 1)y2E

[(y + xATµ

)ν−2]

+ λ

∫ +∞

−∞

{E

[(y + xezATµ

)ν]− E[(

y + xATµ

)ν]}fY (z)dz = 0.

In the special case x = 1 and y = 0, since∫ +∞

−∞

[E

(ezνAν

Tµ

)− E

(Aν

Tµ

)]fY (z)dz = E

(Aν

Tµ

)

m∑

i=1

piηi

ηi − ν+

n∑

j=1

qjθj

θj + ν− 1

,

we obtain that for any ν ∈ (0, β1),σ2

2ν2 + (r − λζ − σ2

2)ν + λ

m∑

i=1

piηi

ηi − ν+

n∑

j=1

qjθj

θj + ν− 1

− µ

E

(Aν

Tµ

)= −νE

(Aν−1

Tµ

),

which is exactly (29) and (19). The proof is completed. ¤

25

Acknowledgements

We would like to thank the Area Editor Mark Broadie, the Associate Editor, two anonymous

referees, Gianluca Fusai, Guillermo Gallego, Paul Glasserman, Vadim Linetsky, Giovanni Pe-

trella, Jan Vecer, Ward Whitt, and participants at INFORMS annual meeting 2009, Workshop

on Optimal Stopping and Singular Stochastic Control Problems in Finance 2009 in Singapore,

Young Researchers Workshop on Finance 2010 in Tokyo, International Symposium on Financial

Engineering and Risk Management 2010 in Taipei, and Columbia-Oxford Risk Summit 2010, for

their helpful comments. The fist author’s research is partially supported by GRF of Hong Kong

RGC (Project No. 610709) and DAG (Project No. DAG11EG07G and DAG08/09.EG07).

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29

Online Supplement

Pricing Asian Options under a Hyper-Exponential Jump Diffusion Model

Ning Cai and S. G. Kou

HKUST and Columbia University

More Discussion on the Numerical Algorithm

1 Stability of the Method

To study the stability of the method, we perform some numerical experiments under the BSM

to show the absolute and relative errors of our double-Laplace inversion method against various

choices of parameters A1, A2 and X. The “true” prices are obtained by using Monte Carlo sim-

ulation with a control variate being∫ t0 eX(s)ds, because it is easy to compute E

[∫ t0 eX(s)ds

]=

(ert−1)/r, and∫ t0 eX(s)ds has a high degree of correlation with the payoff function. In addition,

Richardson extrapolation is also employed to reduce the discretization bias generated when we

discretize the sample path to approximate the integral. More precisely, let M(h) be the Monte

Carlo estimator without Richardson extrapolation when the discretization step size is set to

be h. Then we use (4M(h) − M(2h))/3 rather than M(h) as the final estimator to achieve

the discretization bias reduction. For more details about the technique of control variates and

Richardson extrapolation, see Glasserman [24].

Figure 1 shows how the absolute and relative errors change as A1, A2 and X vary in the

case of low volatility σ = 0.05, illustrating that our algorithm is insensitive to the selection of

parameters A1, A2 and X. For normal volatilities, our method becomes even more stable and

associated plots can be obtained on request.

2 Discretization Error Bounds of Euler Inversion Algorithm un-der the BSM

The discretization error bound of the Euler inversion algorithm was first studied by Abate and

Whitt [1], and was extended to a two-sided Laplace inversion case by Petrella [37]. In this

A-1

30 40 50 60 70−2

−1

0

1

2x 10

−3 σ=0.05

Ab

so

lute

err

ors

Parameter A1

30 40 50 60 70

−1

0

1

x 10−3 σ=0.05

Re

lative

err

ors

Parameter A1

30 35 40 45 50 55 60−2

−1

0

1

2x 10

−3 σ=0.05

Ab

so

lute

err

ors

Parameter A2

30 35 40 45 50 55 60

−1

0

1

x 10−3 σ=0.05

Re

lative

err

ors

Parameter A2

250 300 350 400 450 500 550 600−2

−1

0

1

2x 10

−3 σ=0.05

Ab

so

lute

err

ors

Parameter X250 300 350 400 450 500 550 600

−1

0

1

x 10−3 σ=0.05

Re

lative

err

ors

Parameter X

Upper Limit of the 95% confidence interval

Upper Limit of the 95% confidence interval

Upper Limit of the 95% confidence interval

Lower Limit of the 95% confidence interval

Lower Limit of the 95% confidence interval

Lower Limit of the 95% confidence interval

Figure 1: The stability and accuracy of the algorithm as A1, A2 and X vary in the case of low volatilityσ = 0.05. The default choices for unvarying algorithm parameters are A1 = 28, A2 = 40 and X given by (27).The absolute errors and relative errors are reported on the left and right graphs, respectively. For broad regionsof A1, A2 and X our algorithm appears to be stable and accurate, all within the 95% confidence intervals. Infact, the relative errors are all smaller than 0.02%.

A-2

subsection, by extending the results in Petrella [37], we provide discretization error bounds

of the inversion algorithm for our specific case of Asian option pricing under the BSM. The

discretization error bounds decay exponentially, therefore leading to a fast convergence.

Recall that what we want to invert is L(µ, ν) =∫∞0

∫∞−∞ e−µte−νkf(t, k)dkdt, where f(t, k) =

XE(S0X At − e−k)+. Then we can prove the following theorem for the error bounds.

Theorem 2.1. Suppose t ∈ (0, A12(θ1+θ2)) and k > A2

θ2, for some constant θ2 > 0, where θ1 =

1+ r+σ2/2 > 0 and r = max(r− σ2

2 , 0). Then the discretization error bounds e+d and e−d satisfy

e+d ≤

C+(θ1)1− e−(A1−2θ1t)

{e−A2

1− e−A2+ e−(A1−2θ1t)

}, (32)

e−d ≤ C−(θ1, θ2)1

1− e−(A1−2(θ1+θ2)t)

e−(θ2k−A2)

1− e−(θ2k−A2), (33)

with C+(θ1) := 2S0eθ1t and C−(θ1, θ2) := 2S0e

tσ2θ22/2+[(1+r+σ2)t−1]θ2+θ1t.

Before proving this theorem, we give an example to illustrate how to apply it in real sit-

uations. Consider the case where r = 0.09, σ = 0.2 and t = 1, and we use A1 = 50,

A2 = 40 and θ2 = 20. Then r = 0.07, θ1 = 1.09, t = 1 ∈(0, A1

2θ1

)≡ (0, 1.19), and k =

4 ∈(

A2θ2

,+∞)≡ (2,+∞). Simple algebra yields that C+(θ1) ≈ e6.39 and C−(θ1, θ2) ≈ e16.59.

Plugging them into (32) and (33), we can get discretization error bounds: e+d ≤ 2.53 × 10−15

and e−d ≤ 6.80 × 10−11. Hence, the discretization error for Asian option price is theoretically

no more than 6.80× 10−11 × e−rt/t ≈ 6.22× 10−11.

Proof of Theorem 2.1: First, since the scaling factor X > S0, we have that

f(t, k) = XE

(S0

XAt − e−k

)+

≤ XE

(S0

XAt − S0

Xe−k

)+

= S0E(At − e−k)+,

where k = log( XKt). On the other hand, we can bound At as follows

At =∫ t

0e(r−σ2

2)s+σW (s)ds ≤

∫ t

0exp

(rt + σ max

{0≤s≤t}W (s)

)ds = t exp

{rt + σ max

{0≤s≤t}W (s)

},

A-3

where r := max(r − σ2

2 , 0). Since max{0≤s≤t}W (s)=d|W (t)|, it follows that

f(t, k) ≤ S0E[t exp {rt + σ|W (t)|} − e−k

]+

= S0E[(

t exp {rt + σW (t)} − e−k)

I{W (t)≥0, t exp(rt+σW (t))>e−k}]

+S0E[(

t exp {rt− σW (t)} − e−k)

I{W (t)<0, t exp(rt−σW (t))>e−k}]

≤ S0E[(

t exp {rt + σW (t)} − e−k)

I{t exp(rt+σW (t))>e−k}]

+S0E[(

t exp {rt− σW (t)} − e−k)

I{t exp(rt−σW (t))>e−k}]

= 2S0E[(

t exp {rt + σW (t)} − e−k)

I{t exp(rt+σW (t))>e−k}],

via the symmetric property of standard Brownian motion.

Next, introduce a new measure P such that dPdP = eYt−(r+σ2/2)t, where Yt := rt + σW (t).

Then the change of measure leads to

f(t, k) ≤ 2S0E[t exp{rt + σWt}I{t exp(rt+σW (t))>e−k} × e−Yt+(r+σ2/2)t

]

= 2S0te(r+σ2/2)tP{t exp(Yt) > e−k}

≤ 2S0e(1+r+σ2/2)tP{Yt > −k − log(t)}

= 2S0eθ1tP{Yt > −k − log(t)},

where θ1 := 1 + r + σ2/2 > 0 and the last inequality holds because t < et for any t > 0.

Therefore, when j1 ≥ 0 and j2 ≥ 0, we have

f((2j1 + 1)t, (2j2 + 1)k) ≤ 2S0eθ1(2j1+1)tP{Yt > −(2j2 + 1)k − log((2j1 + 1)t)}

≤ 2S0eθ1te2θ1j1t = C+(θ1)e2θ1j1t,

where C+(θ1) := 2S0eθ1t. On the other hand when j1 ≥ 0 and j2 ≤ −1, we have that for any

θ2 > 0,

f((2j1 + 1)t, (2j2 + 1)k) ≤ 2S0eθ1(2j1+1)tP{Yt > −(2j2 + 1)k − log((2j1 + 1)t)}

≤ 2S0eθ1(2j1+1)tP{Yt > −j2k − log((2j1 + 1)t)}

≤ 2S0eθ1(2j1+1)tE

(eθ2Yt

)eθ2j2k+θ2 log((2j1+1)t),

A-4

where the second inequality holds as j2 ≤ −1 and the third inequality comes from Markov’s

inequality. Since x + 1 ≤ ex for any x > −1, we obtain that eθ2 log((2j1+1)t) ≤ eθ2[(2j1+1)t−1] and

f((2j1 + 1)t, (2j2 + 1)k) ≤ 2S0eθ1(2j1+1)tE

(eθ2Yt

)eθ2j2k+θ2[(2j1+1)t−1]

= 2S0e(θ1+θ2)t−θ2E

(eθ2Yt

)· e2(θ1+θ2)j1teθ2j2k

= C−(θ1, θ2)eθ2j2k+2(θ1+θ2)j1t,

where

C−(θ1, θ2) := 2S0e(θ1+θ2)t−θ2 · E

(eθ2Yt

)= 2S0e

(θ1+θ2)t−θ2 · E(e(θ2+1)Yt−(r+σ2/2)t

).

Recall that Yt = rt + σW (t). Simple algebra yields

C−(θ1, θ2) = 2S0etσ2θ2

2/2+[(1+r+σ2)t−1]θ2+θ1t.

If we have t ∈(0, A1

2θ1

), according to the definition of e+

d and the bound of function f((2j1 +

1)t, (2j2 + 1)k) obtained above, we can get

e+d ≤

∞∑

j2=1

∞∑

j1=0

e−(j1A1+j2A2)C+(θ1)e2θ1j1t +∞∑

j1=1

e−j1A1C+(θ1)e2θ1j1t

= C+(θ1)∞∑

j2=1

∞∑

j1=0

e−(A1−2θ1t)j1−j2A2 + C+(θ1)∞∑

j1=1

e−(A1−2θ1t)j1

= C+(θ1)e−A2

1− e−A2

11− e−(A1−2θ1t)

+ C+(θ1)e−(A1−2θ1t)

1− e−(A1−2θ1t)

=C+(θ1)

1− e−(A1−2θ1t)

{e−A2

1− e−A2+ e−(A1−2θ1t)

},

which is exactly (32).

For e−d we have for any t ∈(0, A1

2(θ1+θ2)

)and k > A2

θ2,

e−d ≤−1∑

j2=−∞

∞∑

j1=0

e−(j1A1+j2A2)C−(θ1, θ2)eθ2j2k+2(θ1+θ2)j1t

= C−(θ1, θ2)∞∑

j1=0

e−(A1−2(θ1+θ2)t)j1

−1∑

j2=−∞ej2(θ2k−A2)

= C−(θ1, θ2)1

1− e−(A1−2(θ1+θ2)t)

e−(θ2k−A2)

1− e−(θ2k−A2),

A-5

from which (33) is proved. ¤

Double-Laplace Inversion Method2 decimal 3 decimal 4 decimal 5 decimal

σ=0.05 (n1, n2)=(35, 35)∗ (35,35) (35,35) (35,35)(CPU time) (3.5 secs)∗∗ (3.5 secs) (3.5 secs) (3.5 secs)

σ=0.1 (n1, n2)=(15;15) (15;35) (15;35) (15;35)(CPU time) (1.2 secs) (2.0 secs) (2.0 secs) (2.0 secs)

Fourier-Laplace Inversion Method2 decimal 3 decimal 4 decimal 5 decimal

σ=0.05 (nl, nf )=(35,115) (35,135) (35,195) (35,195)(CPU time) (17.5 secs) (20.1 secs) (28.6 secs) (28.6 secs)

σ=0.1 (nl, nf )=(15,55) (15,75) (15,95) (15,95)(CPU time) (4.6 secs) (6.4 secs) (7.7 secs) (7.7 secs)

Table 7: Comparison of the efficiency between our double-Laplace inversion and Fusai’s Fourier-Laplace in-version method. In this table, (n1, n2)=(35, 35)∗ means that (n1, n2) should be set roughly at least (35,35) toachieve 2-decimal accuracy. (3.5 secs)∗∗ below (n1, n2)=(35, 35)∗ means that the corresponding CPU time is 3.5seconds. The CPU times associated with Fusai’s method are obtained using the code implemented by Matlab7.1. All computations in Table 7 are conducted on an IBM laptop with a Pentium M 1.86GHz processor. Wecan see that to achieve the same accuracy, our method is more efficient than Fusai’s.

15 20 25 30 35−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Absolute errors of FL prices vs. parameter Al when σ=0.1

Abs

olut

e er

rors

of F

L pr

ices

Parameter Al

15 20 25 30 35−0.5

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Absolute errors of LL prices vs. parameter A1 when σ=0.1

Parameter A1

Figure 2: Comparison of the stability and accuracy between the double-Laplace inversion and the Fourier-Laplace inversion method in the case of low volatility σ = 0.1 under Kou’s model, where other parameters areK = 100, S0 = 100, t = 1, λ = 3, r = 0.09, p1 = 0.6, q1 = 0.4, and η1 = θ1 = 25;. The absolute errors ofLL prices (obtained by the double-Laplace inversion method) and FL prices (obtained by the Fourier-Laplaceinversion method) are reported on the left and right graphs, respectively. Other parameters for the left graphare n1 = 35, n2 = 55, A2 = 40 and X = 5460; while other parameters for the right graph are nl = 35, nf = 135and Af = 40. We can see that LL prices are quite accurate and stable when A1 varies between [22.2,38], but FLprices are so desultory that we cannot decide which Al we should choose.

A-6

3 Comparison with the Fourier and Laplace Inversion Algo-rithm

Under the BSM, Fusai [22] gave a closed-form for the Fourier-Laplace transform of Asian option

price w.r.t. k = ln(σ2Kt/(4S0)) and h = σ2t/4, respectively. Despite some similarities, there

are some key differences between our method and Fusai’s method. (1) Our method performs

better for low volatility, e.g., σ = 0.05 or 0.1. Specifically, for Fusai’s method, a large number

of terms are needed to do the Euler inversion to achieve a desired accuracy. In comparison,

our algorithm in general requires far fewer terms in computation, especially for low volatility

(See Table 7). This is mainly because we use the latest inversion method with a scaling factor

in Petrella [37]. (2) Our method performs better in jump diffusion models. Specifically, the

Fourier-Laplace inversion method seems unstable in the case of low volatilities under Kou’s

model because it is sensitive to parameters. To illustrate the sensitivity, we fix Af = 40 and let

Al change from 15 to 38. The right panel of Figure 2 illustrates how the difference between the

numerical result and the true value changes as Al varies in the case of σ = 0.1 and K = 100.

In comparison with the double-Laplace inversion on the left panel of Figure 2, FL prices seem

unstable. Figure 2 seems to indicate that, with jumps, our double-Laplace inversion method

works in a more stable manner than the Fourier-Laplace method, especially in the case of low

volatility. (3) Under the BSM, we can derive a theoretical discretization error bound for the

double-Laplace inversion. See Section 2 in the online supplement. (4) In terms of the main

theoretical difference, note that the recursion used in Fusai’s paper, namely (8), has no unique

but infinitely many solutions. We spend considerable efforts to overcome this difficulty; see

Section 3.

A-7